\documentclass{article} \voffset=-0.05\textheight \textheight=1.10\textheight \hoffset=-0.05\textwidth \textwidth=1.1\textwidth \include{macros} \title{Welcome to the MAGMA Computer Algebra System} \date{December 3, 2001} \author{William Stein\\ {\tt http://modular.fas.harvard.edu}} \begin{document} \maketitle \tableofcontents \section{Introduction} \magma{} is a sophisticated mathematics calculator that is significantly different than Maple, Mathematica, and PARI/GP. In this talk, which is directed at number theorists interested in modular forms, I will introduce \magma{}, and give you a brief tour of some aspects of \magma{}. \vspace{1ex} \noindent{\bf Acknowledgement:} Allan Steel, Kevin Buzzard, Helena Verrill, and John Cannon provided helpful comments on these notes. \section{MAGMA} \mbox{}\indent{\bf What is M{\small AGMA}, ``the group''?} \magma{} is produced/headed by Professor John Cannon of the University of Sydney, along with two handfulls of Ph.D. students, postdocs, and ``some well-groomed professional top-notch elite C programmers/hackers''. They are nonprofit\footnote{Cost recover is the technical term.}, in the sense that all the money from \magma{} licenses and grants goes towards basic maintenance and flying mathematicians (like me) there. %\magma{} resembles several German groups like LiDIA or KANT, except %that \magma{} is not free and Cannon wears the good manager hat as %well as the mathematician one. {\bf What is M{\small AGMA}, ``the program''?} Unlike Maple and Mathematica, which target college teachers, calculus students, engineers, and scientists, \magma{} is aimed squarely at research mathematicians (and intelligence agencies). My TI-89 pocket calculator is much better at calculus than \magma{}, but \magma{} comes standard with a package by Michael Stoll for arithmetic in Jacobians of Hyperelliptic curves and my package for computing with modular forms. {\bf \magma{} differs from PARI} in that its scope is vastly greater, and \magma{} provides more powerful tools for suporting very large projects. Highly optimized implementation of algebraic number theory algorithms is the reason for existence of PARI. Also, PARI's ``philosophy'' is to assume standard conjectures (e.g., Riemann Hypothesis) in order to speed up computations, but \magma{} typically does not. A {\bf fundamental difference} between \magma{} and the other systems is that every object in \magma{} {\bf belongs to a category}; this is analogous to how variables in {\tt C} must have a type. Thus there are literally hundreds of entirely different and incompatible notions of ``$1$'' in \magma{}. \section{Resources for Learning MAGMA} If you want to really learn \magma{} (or anything!) it's absolutely essential to try it! \magma{} will soon be available for everyone's use on {\tt abel.math.harvard.edu}. (Caveat: If you wish to do very resource-intensive computations, get an account on {\tt modular.fas.harvard.edu} from me.) Type {\tt magma} at the command prompt and try it out! There are many resources to get you going and stay headed in the right direction. See \begin{center} {\tt http://modular.fas.harvard.edu/docs/.} \end{center} \begin{itemize} \item {\tt first.dvi} (16 pages): First few steps in \magma. Gets you going. \item {\tt tutorial.dvi} (11 pages): A {\em Modular Symbols Tutorial} by Kevin Buzzard and myself. Spend 1 hour with this and you'll be able to do many computations with modular symbols in \magma. \item {\tt overv27.dvi} (64 pages): An overview of the domains of mathematics \magma{} treats. This is humbling. \item {\tt examples.dvi} (200 pages): A short book of sophisticated examples. \item {\tt hb[1--8].dvi} (thousands of pages): Reference manual for all functions of \magma{} (also available in HTML). If you want to know what a \magma{} function really does, look here. I wrote the chapters about modular symbols and modular forms, which contain nontrivial examples and some open problems. \item {\tt intro.dvi} (800 pages): A book about how to program in \magma, with sophisticated examples. \item {\tt william\_stein.human}: I have lots of experience with \magma, and I'm human hence more interactive than the above resources. \end{itemize} \section{MAGMA Categories} A first glance at \magma{} reveals nothing too out of the ordinary: \begin{verbatim} > a := 2; > a; 2 > a + 3*a^2; 14 \end{verbatim} However, \begin{verbatim} > f := x^2+1; >> f := x^2+1; ^ User error: Identifier 'x' has not been declared or assigned > \end{verbatim} Here \magma{} doesn't know what~$x$ is, and it doesn't {\em assume it does}. The fix is to tell \magma{} what~$x$ is: \begin{verbatim} > // This is a comment, just like in C++. "//" means comment. > Z := Integers(); > Z; // Shortcut for Print(Z); Integer Ring > R := PolynomialRing(Z); // R is Z[x] > R; Univariate Polynomial Ring in x over Integer Ring > f := x^2 + 1; // OK now > f; x^2 + 1 \end{verbatim} Everything in \magma{} is an object in a certain category and has a parent. \begin{verbatim} > Category(Z); RngInt // one object category > Category(R); RngUPol // the category of univariate polynomial rings > Category(f); RngUPolElt // the category of elements of univariate polynomial rings > Parent(f); Univariate Polynomial Ring in x over Integer Ring // That's R. \end{verbatim} \section{Some Rings} Arithmetic in many {\bf standard rings} is supported in \magma: $$\Z, \quad \Z/N\Z, \quad \F_{q},\quad \Z_p, \text{ and } R[[q]].$$ \begin{verbatim} > S := IntegerRing(12); // make Z/12Z > S; Residue class ring of integers modulo 12 > Set(S); // the forgetful functor to SetEnum { 0, 11, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 } > W := PowerSeriesRing(S); // W is S[[q]] > W; Power series ring in q over IntegerRing(12) > 1/(1+q); // arithmetic in S[[q]]. 1 + 11*q + q^2 + 11*q^3 + q^4 + 11*q^5 + O(q^6) \end{verbatim} \section{Finite Fields} Support for basic arithmetic over finite fields is a dream because a choice of $\Fbar_p$ is built in. There is a fixed compatible choice of maps $\F_{p^n}\hookrightarrow \F_{p^{nm}}$ for all $n,m\geq 1$. \begin{verbatim} > K := GF(3^2); > MinimalPolynomial(a); $.1^2 + 2*$.1 + 2 > // The $.1 means "the first unnamed generator". $.2 would be the second, etc. > L := GF(3^6); // b generates L^* > L!a; // image of a in L. b^91 // sweet. > A := MatrixAlgebra(L,2)![b,2,0,1]; > A; [ b 2] [ 0 1] > V := VectorSpace(L,2); > v := V.1; > v; ( 1 0) > v*A; // WARNING: In Australia, matrices act ON THE RIGHT! ( b 2) \end{verbatim} %> G := GeneralLinearGroup(2,K); // build structures upon structures %> #G; %5760 %> Generators(G); %{ %[ 2 1] %[ 2 0], % %[ a 0] %[ 0 1] %} \section{Commutative Algebra and Geometry} It is easy and natural to work with quotients of polynomial rings and ideals in MAGMA. \begin{verbatim} > // The field K = Q(a,b,x0,x1,x2) > K := FieldOfFractions(PolynomialRing(Rationals(),5)); Rational function field of rank 5 over Rational Field Variables: a, b, x0, x1, x2 > // The polynomial ring R = K[y0,y1,y2] > R := PolynomialRing(K,3); Polynomial ring of rank 3 over Rational function field of rank 5 over Rational Field [...] > // A maximal ideal of R. > I := ideal; \end{verbatim} The quotient $L = R/I$ contains three generic points on the elliptic curve $y^2 = x^3 + ax + b$. \begin{verbatim} > L := quo; Affine Algebra of rank 3 over [...] > E := EllipticCurve([L| a,b]); Elliptic Curve defined by y^2 = x^3 + a*x + b over Affine Algebra of rank 3 over Rational function field of rank 5 over Rational Field Variables: a, b, x0, x1, x2 > P0 := E![L|x0,y0]; P1 := E![L|x1,y1]; P2 := E![L|x2,y2]; > P0 + P1; (-2/(x0^2 - 2*x0*x1 + x1^2)*y0*y1 + ... > // left and right are huge -- you probably don't want to print them out. > left := (P0 + P1) + P2; right := P0 + (P1 + P2); > left eq right; true \end{verbatim} \section{Unique Flexibility} This is a short example which demonstrates the flexibility of the system. \begin{verbatim} > R := PolynomialRing(Integers()); > f := x^3+1; > Factorization(f); [ , ] > S := PolynomialRing(pAdicField(7)); > Factorization(S!f); [ , <(1 + O(7^20))*x - 22143577275619761 + O(7^20), 1>, <(1 + O(7^20))*x + 22143577275619760 + O(7^20), 1> ] > W := PolynomialRing(GF(7)); > Factorization(W!f); [ , , ] > F := GF(5^2); > R5 := PolynomialRing(F); > Factorization(R5!f); [ , , ] \end{verbatim} \section{Some Frequently Asked Questions} \begin{question} Is there an equivalent to a {\tt .magmarc} where initial commands can be loaded? (e.g., setting {\tt vi} or {\tt emacs} mode) \end{question} Yes. Set the environment variable {\tt MAGMA\_STARTUP\_FILE} to the file you want \magma{} to execute at startup. Also, you should learn about {\tt spec} files. Check out \begin{verbatim} http://modular.fas.harvard.edu/docs/magma/htmlhelp/text122.html#247 \end{verbatim} \begin{question} When running magma I tend to have a separate window with the documentation available, but also sometimes type {\tt ?X} for some {\tt X}; sometimes the output of available options is more than a screen long; is there a way to make a paginator (e.g., less) do this automatically, a page at a time? \end{question} I {\em never} use {\tt ?X} for help. Use the HTML help, which comes with \magma{}. You can also use it by viewing: \begin{verbatim} http://modular.fas.harvard.edu/docs/magma/htmlhelp/MAGMA.html \end{verbatim} This HTML help is generated from exactly the same \TeX{} source files as the {\tt ?} help, but by using the index it is easier to view. \begin{question} How can one refer to the output of the most recent command? \end{question} \begin{verbatim} $ and $1, $2, $3. \end{verbatim} \begin{question} What are the easy ways for storing output in a file? \end{question} Here's an example. \begin{verbatim} > file := Open("testfile","w"); > fprintf file, "hello world"; > Flush(file); > r := Open("testfile","r"); > Gets(r); hello world \end{verbatim} You can read about file I/O commands at \begin{verbatim} http://modular.fas.harvard.edu/docs/magma/htmlhelp/text130.html#363 \end{verbatim} \begin{question} Is there a simple way to take an polynomial in $x$ and replace $x$ by some other expression? \end{question} No, because that doesn't make \magma{}matically rigorous sense in general. However, in cases where what you want to do does make sense (which should always be the case, I trust), you can use the {\tt hom} construction. Here is an example: \begin{verbatim} > R := PolynomialRing(RationalField()); > phi := hom< R -> RationalField() | 4>; > phi(x^2+1); 17 \end{verbatim} \end{document}